L(s) = 1 | + (−1.15 + 1.15i)3-s + (2.51 − 2.51i)5-s + 4.37i·7-s + 0.313i·9-s + (−3.31 + 0.130i)11-s + (−2.19 + 2.19i)13-s + 5.82i·15-s − 0.470i·17-s + (−0.632 − 0.632i)19-s + (−5.07 − 5.07i)21-s − 6.62·23-s − 7.61i·25-s + (−3.84 − 3.84i)27-s + (−0.0549 + 0.0549i)29-s + 0.184i·31-s + ⋯ |
L(s) = 1 | + (−0.669 + 0.669i)3-s + (1.12 − 1.12i)5-s + 1.65i·7-s + 0.104i·9-s + (−0.999 + 0.0393i)11-s + (−0.609 + 0.609i)13-s + 1.50i·15-s − 0.114i·17-s + (−0.145 − 0.145i)19-s + (−1.10 − 1.10i)21-s − 1.38·23-s − 1.52i·25-s + (−0.739 − 0.739i)27-s + (−0.0102 + 0.0102i)29-s + 0.0331i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.192i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6504377470\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6504377470\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + (3.31 - 0.130i)T \) |
good | 3 | \( 1 + (1.15 - 1.15i)T - 3iT^{2} \) |
| 5 | \( 1 + (-2.51 + 2.51i)T - 5iT^{2} \) |
| 7 | \( 1 - 4.37iT - 7T^{2} \) |
| 13 | \( 1 + (2.19 - 2.19i)T - 13iT^{2} \) |
| 17 | \( 1 + 0.470iT - 17T^{2} \) |
| 19 | \( 1 + (0.632 + 0.632i)T + 19iT^{2} \) |
| 23 | \( 1 + 6.62T + 23T^{2} \) |
| 29 | \( 1 + (0.0549 - 0.0549i)T - 29iT^{2} \) |
| 31 | \( 1 - 0.184iT - 31T^{2} \) |
| 37 | \( 1 + (3.55 - 3.55i)T - 37iT^{2} \) |
| 41 | \( 1 - 3.77T + 41T^{2} \) |
| 43 | \( 1 + (-4.82 + 4.82i)T - 43iT^{2} \) |
| 47 | \( 1 + 0.258iT - 47T^{2} \) |
| 53 | \( 1 + (5.27 - 5.27i)T - 53iT^{2} \) |
| 59 | \( 1 + (-1.64 - 1.64i)T + 59iT^{2} \) |
| 61 | \( 1 + (9.05 - 9.05i)T - 61iT^{2} \) |
| 67 | \( 1 + (1.83 - 1.83i)T - 67iT^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 1.80T + 79T^{2} \) |
| 83 | \( 1 + (8.61 + 8.61i)T + 83iT^{2} \) |
| 89 | \( 1 + 14.8iT - 89T^{2} \) |
| 97 | \( 1 + 9.13T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.900156767130795178484296974233, −9.182379869321908811580539227701, −8.632368934720684850015941003524, −7.67325371245069764265661364294, −6.12858487244138033752132547040, −5.67496477715914504351444710782, −5.06919143258309035331213719388, −4.44354075863664054929190780305, −2.56959482512589604338719243421, −1.90163899193552335997898757252,
0.26168129366855675831137924855, 1.71108756537989230322167093228, 2.85130920901725258489155441373, 3.94114959236865649260506400877, 5.25021705846912243340867949209, 6.07242112049273884135203150490, 6.67642863378061894463314130257, 7.43105546533549836621949054346, 7.945506782845948305033440101852, 9.618792980739799168997875159664